do we have $:\left\Vert Q^{n+1}x\right\Vert \leq \varepsilon \left\Vert Q^{n}x\right\Vert $ for all $x\in \mathcal{H}$ for a quasi-nilpotent operator?

Question

Let $\mathcal{H}$ be a Hilbert space and let $Q$ be a bounded quasi-nilpotent operator on $\mathcal{H}$.

I'm trying to prove that for every$\ \varepsilon >0,$ there is some $n\in %TCIMACRO{\U{2115} }% %BeginExpansion \mathbb{N} %EndExpansion ,$ such that $:\left\Vert Q^{n+1}x\right\Vert \leq \varepsilon \left\Vert Q^{n}x\right\Vert $ for all $x\in \mathcal{H}$.

Thank you !

Answer 1

Could this operator be a counter example? Define $T$ on $H=l^2$ as $$ Tx =(0, \frac{x_1}{2^1}, x_2, \frac{x_3}{2^3}, x_4, \dots ). $$